Assume that we have a sequence of $n$ realizations $x_1, x_2, ..., x_n$ of an i.i.d random variable $X$ with cdf $F_X(x)$ and pdf $f_X(x)$. Now define $Y$ as
$Y_i = \sum_{j=i}^{k} x_j, \forall i \in \{1,...,n-k+1\}$
which generates the sequence $Y_1, ..., Y_{n-k+1}$ of dependent random variables as $Y_i$ contains $k-1$ elements of $Y_{i-1}$ (and $Y_{i+1}$).
Now I want to find the covariance $cov(Y_i, Y_j), \ i \neq j$ and the covariance matrix of $Y$. As the covariance is dependent on the initial realizations $x_1, x_2, ..., x_n$ I am struggeling to get a general formulation.
What is the correct formulation for $cov(Y_i, Y_j)$ and $\sum_Y$?